Understanding the Averaging of Exam Scores: A Detailed Example

In this lesson, we delve into an interesting problem that involves the calculation of exam scores for a group of students. Specifically, we explore a scenario where six students write an exam, and we are given certain details to determine the score of the sixth student based on the average score of the group.

Given Information

We are told that the average score obtained by six students on their exam is 71. Two of these students each obtained a score of 73, and the next three students each obtained a score of 68. Our task is to find the score of the sixth student.

Calculation Steps

Total Marks Calculation

First, we will calculate the total marks obtained by all six students.

The total marks of six students 6 times; 71 426 The total marks of two students 2 times; 73 146 The total marks of three students 3 times; 68 204

The total marks of five students 146 204 350

Hence, the score of the sixth student 426 – 350 76

Algebraic Approach

We can also approach this problem using an algebraic equation. Let's denote the score of the sixth student by x.

The average score of six students is 71, so we have:

$frac{2 times 73 3 times 68 x}{6} 71$

Multiplying both sides by 6, we get:

2 times; 73 3 times; 68 x 6 times; 71

Simplifying the left-hand side:

146 204 x 426

x 426 - 350

x 76

Verification Using Mathematical Formula

We can verify our answer using the formula for the average. The average score of 6 students is 71, so:

6 times; 71 426 (total score for 6 students)

The total score for the first 4 students (two with 73 and three with 68) is 146 204 350.

Hence, the score of the sixth student must be 426 - 350 76.

General Conclusion

The score of the sixth student is 76. This problem demonstrates the application of algebraic methods in solving real-world problems involving averages and finding missing values.

Key Concepts and Vocabulary

This example reinforces the concepts of:

Average Score: Calculating the total score and dividing by the number of students. Marks and Scores: Understanding individual and group scores in exams. Algebraic Problem Solving: Using equations to find unknown values.

Practice Problem

Try solving a similar problem:

Problem: Seven students write an exam. The average score is 82. Three students each score 85 and two score 78. Find the score of the remaining two students.

Solution: The total score for seven students is 7 times; 82 574. The total score for three students is 3 times; 85 255, and for two students is 2 times; 78 156. Hence, the combined score of the remaining two students must be 574 - (255 156) 163.

Conclusion

Understanding how to calculate and work with averages is a crucial skill in mathematics and can be applied in various practical scenarios. By using algebraic methods, we can effectively determine unknown values based on given averages and total scores.

If you found this explanation helpful, feel free to explore more complex problems or seek additional resources to enhance your grasp on mathematics and problem-solving techniques.